00000310.htm

6 LINEAR DIFFERENTIAL EQUATIONS 293

\\f(t_yx_i)-f(t,x₂)\\^k(\t-t₀\)\\x_l-x₂\\ for re I, x_l9x₂ in E, where
£-*&(£) is a regulated function in [0, r[. Then for every x₀ e E, there exists a
unique solution u of (10.4.1), defined in I, and such that u(t₀) = x₀.

We only have to prove that, if c is the l.u.b. of the numbers p such that
0 < p < r and that there exists a solution of (10.4.1) defined in \t — t₀\ < p
and taking the value X_Q at r₀, then c = r (by (10.5.4)). Suppose the contrary;
then, by (10.5.4), there is a solution v of (10.4.1) defined in J :\t — t₀\<c
and such that v(t₀) = X_Q . We are going to show that the conditions of (10.5.5)
are satisfied; applying (10.5.5) then yields a contradiction and ends the proof.

As here H = E, the condition v(J) c: H is trivially verified, so we have only
to check that t ->/(f, v(t)) is bounded in J. Now, in the compact interval [0, c], k
is bounded and so is the continuous function /-> ||/(r, jc₀)|| in the compact
set J; hence there exist two numbers m > 0, h > 0 such that \\f(t, x)\\ ^
m\\x\\ + h for t e J and x e E. This implies \\v'(t)\\ ^ w||»(r)|| + h for t e J; if
we write w(£) = ||i>(r₀ + ^£)ll with |/l| = 1, the mean value theorem shows that
w(£) ^ \\XQ\\ + he •+- ml w(Q d£. We therefore can apply Gronwall's lemma
(10.5.1.3), which shows that ||i?(f)|| < ae^mlt~^to{ + b in J (a and b constants),
hence v is bounded in J, and so is ||/(r, i?(f))|| < /w||»(OII + h.

(10.6.1.1) Here again, when K = R, the condition of continuity on/can be
relaxed to condition (a) of Remark (10.4.6).

A linear differential equation is an equation (10.4.1) of the special form
(1 0.6.2) x' = A(t) - x + b(t) ( =f(t, x))

where A is a mapping of I into the Banach space JSf(E; E) of continuous
linear mappings of E into itself (Section 5.7), and b a mapping of I into E. We
have here H = E, and by (5.7.4)

for all re I, x_l9 x₂ in E. Applying (10.6.1) and (10.6.1.1) we therefore get:

(1 0.6.3) Let I c K be an open ball of center t_Q . Suppose A and b are regulated
in I if K = R, analytic in I if K = C. Then, for every X_Q e E, there exists a
unique solution u of (10.6.2), defined in I and such that u(t₀) = x₀ .

Observe that if b = 0, and x₀ = 0, the solution u of (10.6.2) is equal to 0.
From (10.6.3) we easily deduce the apparently more general result:



	6 LINEAR DIFFERENTIAL EQUATIONS 293 \\f(t_yx_i)-f(t,x₂)\\^k(\t-t₀\)\\x_l-x₂\\ for re I, x_l9x₂ in E, where £-&(£) is a regulated function in* [0, r[. Then for every x₀ e E, there exists a unique solution u of (10.4.1), defined in I, and such that u(t₀) = x₀. We only have to prove that, if c is the l.u.b. of the numbers p such that 0 < p < r and that there exists a solution of (10.4.1) defined in \t — t₀\ < p and taking the value X_Q at r₀, then c = r (by (10.5.4)). Suppose the contrary; then, by (10.5.4), there is a solution v of (10.4.1) defined in J :\t — t₀\<c and such that v(t₀) = X_Q . We are going to show that the conditions of (10.5.5) are satisfied; applying (10.5.5) then yields a contradiction and ends the proof. As here H = E, the condition v(J) c: H is trivially verified, so we have only to check that t ->/(f, v(t)) is bounded in J. Now, in the compact interval [0, c], k is bounded and so is the continuous function /-> \|\|/(r, jc₀)\|\| in the compact set J; hence there exist two numbers m > 0, h > 0 such that \\f(t, x)\\ ^ m\\x\\ + h for t e J and x e E. This implies \\v'(t)\\ ^ w\|\|»(r)\|\| + h for t e J; if we write w(£) = \|\|i>(r₀ + ^£)ll with \|/l\| = 1, the mean value theorem shows that w(£) ^ \\XQ\\ + he •+- ml w(Q d£. We therefore can apply Gronwall's lemma (10.5.1.3), which shows that \|\|i?(f)\|\| < ae^mlt~^to{ + b in J (a and b constants), hence v is bounded in J, and so is \|\|/(r, i?(f))\|\| < /w\|\|»(OII + h.

	(10.6.1.1) Here again, when K = R, the condition of continuity on/can be relaxed to condition (a) of Remark (10.4.6). A linear differential equation is an equation (10.4.1) of the special form (1 0.6.2) x' = A(t) - x + b(t) ( =f(t, x)) where A is a mapping of I into the Banach space JSf(E; E) of continuous linear mappings of E into itself (Section 5.7), and b a mapping of I into E. We have here H = E, and by (5.7.4)

	for all re I, x_l9 x₂ in E. Applying (10.6.1) and (10.6.1.1) we therefore get: (1 0.6.3) Let I c K be an open ball of center t_Q . Suppose A and b are regulated in I if K = R, analytic in I if K = C. Then, for every X_Q e E, there exists a unique solution u of (10.6.2), defined in I and such that u(t₀) = x₀ . Observe that if b = 0, and x₀ = 0, the solution u of (10.6.2) is equal to 0. From (10.6.3) we easily deduce the apparently more general result: